lec0202 - CS 173: Discrete Mathematical Structures Cinda...

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CS 173: Discrete Mathematical Structures Cinda Heeren heeren@cs.uiuc.edu Rm 2213 Siebel Center Office Hours: W 9:30-11:30a
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Cs173 - Spring 2004 CS 173 Announcements Homework 3 available. Due 02/ 05, 8a. Midterm 1: 2/ 23/ 06, 7-9p, location TBA. Let’s try the remotes… For breakfast I ate: a) cereal b) eggs c) brew d) donuts
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Cs173 - Spring 2004 CS 173 Set Theory - A proof for us to do together. X (Y - Z) = (X Y) - (X Z). True or False? Prove your response. = (X Y) (X’ U Z’) = (X Y X’) U (X Y Z’) = U (X Y Z’) = (X Y Z’) (X Y) - (X Z) = (X Y) (X Z)’ = X (Y - Z)
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Cs173 - Spring 2004 Suppose to the contrary, that A B , and that x A B. CS 173 Set Theory - A proof for us to do together. Pv that if (A - B) U (B - A) = (A U B) then ______ Then x cannot be in A-B and x cannot be in B-A. Do you see the contradiction yet? But x is in A U B since (A B) (A U B). B = Thus, A . a) A U B = b) A = B c) d) A-B = B-A = Then x is not in (A - B) U (B - A). DeMorgan’s!! Trying to pv p --> q Assume p and not q, and find a contradiction. Our contradiction was that sets weren’t equal.
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Cs173 - Spring 2004 CS 173 Set Theory - Generalized Union A i i =1 n U = A 1 A 2 K A n Ex. Let U = N , and define: A i = { x : $ k >1, x = ki , k N} A 1 = {2,3,4,…} A 2 = {4,6,8,…} A 3 = {6,9,12,…}
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Cs173 - Spring 2004 CS 173 Set Theory - Generalized Union A i i =1 n U = A 1 A 2 K A n Ex. Let U = N , and define: A i = { x : $ k >1, x = ki , k N} Then A i i =2 U a) Primes b) Composites α29 a) N b) I have no clue. primes
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Cs173 - Spring 2004 CS 173 Set Theory - Generalized Intersection A i i =1 n I = A 1 A 2 K A n Ex. Let U = N , and define: A i = { x : $ k , x = ki , k N} A 1 = {1,2,3,4,…} A 2 = {2,4,6,…} A 3 = {3,6,9,…}
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Cs173 - Spring 2004 CS 173 Set Theory - Generalized Intersection A i i =1 n I = A 1 A 2 K A n Ex. Let U = N , and define: A i = { x : $ k , x = ki , k N} Then A i i =1 n I = ? Multiples of LCM(1,…,n)
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Cs173 - Spring 2004 CS 173 Set Theory - Inclusion/Exclusion Example: How many people are wearing a watch?
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lec0202 - CS 173: Discrete Mathematical Structures Cinda...

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